Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

TL;DR

This paper investigates the equivalence problem for minimal rational curves on uniruled projective manifolds, focusing on cases where the varieties of minimal rational tangents are isotrivial, and identifies conditions under which they are locally equivalent to a flat model.

Contribution

It establishes criteria for when families of minimal rational curves with isotrivial varieties of minimal rational tangents are locally equivalent to the flat model, especially for certain hypersurfaces.

Findings

Conditions for isotrivial varieties of minimal rational tangents to be locally equivalent to the flat model.

Application of criteria to non-singular hypersurfaces of degree ≥ 4.

Extension of Cartan's equivalence problem to minimal rational curves.

Abstract

We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when $Z$ satisfies certain projective-geometric conditions, which hold for a non-singular hypersurface of degree $\geq 4$.